Question: You have found the following ages (in years) of 6 sloths. Those sloths were randomly selected from the 23 sloths at your local zoo: $ 14,\enspace 21,\enspace 11,\enspace 10,\enspace 2,\enspace 18$ Based on your sample, what is the average age of the sloths? What is the standard deviation? You may round your answers to the nearest tenth.
Solution: Because we only have data for a small sample of the 23 sloths, we are only able to estimate the population mean and standard deviation by finding the sample mean $({\overline{x}})$ and sample standard deviation $({s})$ To find the sample mean , add up the values of all $6$ samples and divide by $6$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6}} $ $ {\overline{x}} = \dfrac{14 + 21 + 11 + 10 + 2 + 18}{{6}} = {12.7\text{ years old}} $ Find the squared deviations from the mean for each sample. Since we don't know the population mean, estimate the mean by using the sample mean we just calculated {1.69} + {68.89} + {2.89} + {7.29} + {114.49} + {28.09}} {{6 - 1}} $ {s^2} = \dfrac{{223.34}}{{5}} = {44.67\text{ years}^2} $ As you might guess from the notation, the sample standard deviation $({s})$ is found by taking the square root of the sample variance $({s^2})$ ${s} = \sqrt{{s^2}}$ $ {s} = \sqrt{{44.67\text{ years}^2}} = {6.7\text{ years}} $ We can estimate that the average sloth at the zoo is 12.7 years old. There is also a standard deviation of 6.7 years.